Problem: $f(r, \theta) = r\sin(\theta)$ What is the partial derivative of $f$ with respect to $\theta$ ? Choose 1 answer: Choose 1 answer: (Choice A) A $r\cos(\theta)$ (Choice B) B $\sin(\theta) + r\cos(\theta)$ (Choice C) C $\sin(\theta)$ (Choice D) D $\cos(\theta)$
Solution: Taking a partial derivative with respect to $\theta$ means treating $r$ like a constant, then taking a normal derivative. $\begin{aligned} \dfrac{\partial f}{\partial \theta} &= \dfrac{\partial}{\partial \theta} \left[ r\sin({\theta}) \right] \\ \\ &= r\cos({\theta}) \end{aligned}$ In conclusion, $\dfrac{\partial f}{\partial \theta} = r\cos(\theta)$